Stability Enhancement in DC Distribution Systems With Constant Power Controlled Converters

Samuele Grillo*, Vincenzo Musolino t, Giorgio Sulligoi+, Enrico Tironi*

*Dipartimento di Elettrotecnica, Politecnico di Milano piazza L. da Vinci 32, 20133 Milano, Italy

Email: {samuele.grillo.enrico.tironi}@polimi.it tDIMAC RED via Giovanni XXIII 25, 20853 Biassono (MB), Italy Email: v.musolino@dimacred.it

+Dipartimento di Ingegneria Industriale e dell'Informazione, Universita degli Studi di Trieste via Valerio 10, 34127 Trieste, Italy Email: giorgio.sulligoi@di3.units.it

Abstract-DC distribution is becoming more and more a suitable technical solution for medium voltage (MV) distribution systems especially in presence of storage devices and renewable energy sources. Among the different challenging issues concern­ing the new topological configurations of DC systems, stability represents a topic of paramount importance.

The paper presents, by means of numerical continuation method, the stability analysis of two DC distribution systems which supply constant power loads (CPLs) connected to the AC grid with an interface converter. CPLs behavior may provoke voltage instability of the DC systems. The paper presents the approach used to maximize, by means of a feedback action on the interface converter, the stable region of operation of the DC distribution systems.

Index Terms-Stability analysis, DC distribution, Hopf bifur­cation, continuation method.

I. I NTRODUCTION

DC for electric systems is becoming more and more a prominent solution to implement innovative functionalities for the grid management when efficiency, renewable energy sources (RESs) integration, storage system integration and easy system reconfiguration are key aspects.

Until now the main experiences for DC system are related on High Voltage Direct Current (HVDC) transmission systems [1]-[ 4], but the interest is growing also in medium and low voltage applications. Especially in telecommunication [5], [6], automotive [7], [8], aircraft [9], [10] and marine [11]-[15] applications and smart grid [16]-[18] applications (in low or medium voltage), the DC technology is becoming prominent for the dissemination of storage technologies and distributed energy sources in order to optimally design the electric sys­tem. The loads, in many industrial applications, typically are directly supplied in DC and also when the supply is AC, for example for huge electric drives in shipboard applications, it is not at constant frequency in order to optimally control the motors, so that typically a stage in DC is required. Also many RESs, i.e., the photo voltaic systems, supply energy directly in DC, as many storage systems are electrochemical devices directly in DC. From this it is evident that the presence of many DC devices, both from the generation side that the load side, can be optimally managed in DC thanks to

978-1-4673-1943-0/12/$31.00 ©2012 IEEE 848

the presence of reliable and cost effective electronic power converters [19]; from this viewpoint the AC stage can be regarded as an additional con-version stage that affects the total system efficiency and reliability.

The DC architecture integration in electric power system is relatively new, especially for the issues concerning the system fault protection [20], [21], circuit breaker coordination [22], [23] and system stability [11], [12], [16], [24]-[26].

In the following a stability study of a DC distribution system is carried out. A methodology is defined to assess system stability in presence of power electronics converters controlled as constant power loads (CPLs). In a CPL the current absorbed increases when the input voltage decreases and this behavior can be modeled as a negative incremental resistance that may produce a destabilizing effect in the system under consideration. Section II presents the models of two medium voltage direct current (MVDC) distribution system layouts feeding CPLs. The effects on system stability of CPLs and a methodology for the system analysis by means of the continuation method are introduced in Section III. In Section IV the numerical simulations of the system in the time domain are reported. Finally conclusion and remarks are reported in Section V.

II. DC LINK MODEL

Fig. 1 shows the well-known circuit model of the DC link considered for stability analysis [27]. In this circuit voltage Vi represents the rectified output of a generator connected to the DC bus. DC bus voltage V directly feeds a constant power load (P).

Figure 1. Circuit model of the electric propulsion system considered for the stability analysis.

The model is a non-linear second order system, in which Land R include the inductance and the resistance of the synchronous machine and of the output rectifier, C models the equivalent capacitance of the output rectifier and of the input inverter and, finally, iL models the constant power controlled converter. Thus, the equations describing the system dynamics are:

(1)

The numerical values of the circuit parameters indicated in Fig. 1 and Fig. 2 are collected in Table I.

TABLE I SYSTEM PA RA METERS VA LUES FOR THE REFERENCE SIMULATION

ACCORDING TO THE CIRCUIT OF FIG . 2 (R = Rl. C = Cl. L = Ll, V = VI A ND i = il FOR THE CIRCUIT SHOWN IN FIG . I).

V Pn Rl R2 Ll L2 C1 C2

V kW 0 0 mH mH � � 400 3.7 4.58 4.58 13.9 0.2 51.4 51.4

III. STABILITY ANALY SIS As reported in [14] and [27] the presence of controlled

constant power load introduces a destabilizing effect that, with a small signal analysis, can be modeled as a negative resistance. In order to generalize the study to the case in which more than one constant power converter is supplied by the same DC link the electric circuit in Fig. 1 is modified as reported in Fig. 2. In this case an additional branch, consisting of the elements R2, L2, C2, and iL2 ' is connected to the DC link at points A, B. In this configuration the system topology may require to feed the second branch through a dedicated DC breaker in order to realize a reliable distribution. In particular as reported in [27]-[29] in realizing a reliable, safe and good­performing DC breaker an important role is played by the limiting inductance. Consequently, in the equivalent model reported in Fig. 2, the inductive element L2 has been added to take into account the equivalent inductance of the DC breaker.

A. One constant-power-load system

---,--­'I

R2 L2

Figure 2. Circuit model of the electric propulsion system considered for the stability analysis in case of two constant power loads supplied by the same DC link.

R2 represents the longitudinal resistance of the second branch feeder while C2 represents the capacity input filter of the load.

In this case the dynamic equations describing the system are:

(2)

where P represents the total power requested by the two loads and k a power sharing coefficient between the two loads.

849

The stability analysis of the circuit shown in Fig. 1 is carried out by means of the numerical continuation method [30]. The complete non-linear differential equations shown in (1) can be studied while varying the parameter P. Fig. 3 shows the results of this analysis. It can be seen that a Hopf bifurcation (H)-a subcritical Hopf bifurcation since the first Lyapunov coefficient is positive-appears at P = 0.629 pu

[31], [32]. This means that before this critical value there exists

0.9

0.8

0.7

0.6 0' E: 0.5

0.'

0.3

0.2

0.1

0 0 0.5 1.5 p [pu]

LP

2.5

Figure 3. Equilibrium curve of the one constant-power load system (I).

an unstable limit cycle surrounding an equilibrium point as that depicted in Fig. 4. If the initial conditions of the state variables v and i fall inside the region delimited by the limit cycle­whose area is function of P-then the stable equilibrium point, marked by the red point in Fig. 4, is reached. On the contrary, if the initial conditions fall outside the limit cycle no steady state equilibrium point can be reached. The basin of attraction of the stable equilibrium point progressively shrinks while increasing P and eventually collapses making unstable the previously-stable equilibrium point. This also means that the portion of the curve from H to LP in Fig. 3 represents the "trajectory" in the phase space of an unstable equilibrium point.

In [14] the problem of extending the basin of attraction and delaying the occurrence of the Hopf bifurcation has been approached by adding, through a feedback path, two signals­proportional to the state variables-to the input DC voltage Vi,

thus making the system (1) become:

{d' L d� = Vi - (R + ki) i - (1 + kv) V

dv P

controlled system with these two values for the control gains is shown in Fig. 5. It can be clearly noticed that the Ropf

(3) bifurcation (R) occurs at a higher value of P (0.774 pu).

-1

-2

C- = i--dt V

:5�.5�--�--�0�.5�--�1�--�1�.�5----�--�2.5 v [pu]

Figure 4. Representation of the unstable limit cycles and of the basins of attraction of the stable equilibrium points of the one constant-power load system described in (1). The limit cycles (blue lines) and the equilibrium points (red dot) are calculated for P = 0.1 pu and P = 0.4 pu.

The values for the two gains ki and kv chosen in [14] are 0.2219 and -0.0321 respectively. The stability analysis of the

0.9

O.B

0.7

0.6

0.5

0.'

0.5

O.B

0.7

0.6

0.5

0.'

0.3

0.2

1.5 p [pu]

(a)

1.5 p [pu]

(c)

2.5

2.5

O.B

� 0.6 LP

0.'

0.2

° 0��0.�' -- 0� .�2--0� .�3--0� .'--�0�.5--�0.�6--�0.-7�0.B p [pu]

Figure 5. Equilibrium curve of the one constant-power load system (3).

B. Two constant-power-loads system

The same stability analysis has been carried out for the system shown in Fig. 2 and described by the system of non­linear differential equations (2). In this system there are two free parameters P and k, i.e., the total power requested by the two constant-power loads and the sharing coefficient of

1.3

1.2

1.1

O.B

0.7

O. 40

L-----------,-0�. 5,----------�--------�1. 5

p [pu] (b)

1.3

0.5 1.5 p [pu] (d)

Figure 6. Equilibrium curves for the two-con stant-power systems without, (a)-(b), and with, (c)-(d), regulation. Each curve refers to a particular value of k E [O,lJ with 0.1 sample step. For plots (c) and (d) the optimal values for gains ki and kv (i.e., 0.1 and -0.18 respectively) have been used.

850

the total requested power between the two loads. Thus the analysis has been carried out sampling the closed interval [0,1], i.e., the domain in which k spans, with a step of 0.1 and letting P increase to detect possible unstable behaviors. The results are shown in Fig. 6a and Fig. 6b in which each equilibrium curve is associated to a value of k as indicated by the arrows. A graphical representation of the limit cycles is not straightforward, since now there are four state variables. It can be noticed that for small k-i.e., when the load is concentrated at the derived branch-despite the fact that C2 increases the overall stability region, Hopf bifurcations still occur for P < 1 pu.

In order to enlarge this region the same approach described in [14] has been used. The control action, proportional to VI and il through the two gains kv and ki, has been added to V(

(4)

In order to find the optimal value of the two control gains

(5)

where f is the curve that describes the trajectory of the projection of the stability limit point on P with respect to k, ki and kv. This calculation could have been performed in analytical way, had a closed form of the inverse of problem (4) been available. Performing this inversion is heavy from a computational viewpoint since it involves the calculation of the limit value of P, as function of k, ki and kv, that makes two eigenvalues of the linearization of (4) pass through the imaginary axis. Thus, a full factorial experiment has been set up in order to evaluate the values of P that satisfy the aforementioned conditions. The control gains ki and kv have been set to span in the set

where KI = {0.05, 0.06, ... ,0.3 } c <Q K2 = { -0.3, -0.29, ... ,0.8 } c <Q

(7)

The results obtained from this analysis are shown in Fig. 7. The values of P above which the two constant-power-Ioads system losses stability for a certain value of k are drawn. In order to make a more effective comparison the values of the limit powers, shown in Fig. 3 and Fig. 5, are reported. These values do not depend on k and are drawn only to set a benchmark. The other three lines represent the limit curves for the system without regulation (black curve) and

851

1.6

1.4

0.8

0.6

O.4 L_--------

--ol_wor - - - ol_wr .......... tl_wor -.- tl_wr_opt _____ tl_wr_wrst -,;

o . 2 O'------:-O

�. 2,------:O� . 4,------:O� . -=-6 --O�.-=-8 --�

k

Figure 7. Comparison of the stability limit points for the two systems and for different values of the control gains (01: one load; tl: two loads; wor: without regulator; wr: with regulator; opt: optimal allocation in K for control gains ki and kv; wrst: worst possible assignment in K for the control gains). The two lines without markers refer to the one constant-power-Ioad system without control (blue solid) and with control (red dashed). Obviously they do not depend on k and they have been represented with the other curves to set a benchmark. The other three lines refer to the two constant-power loads system: without control (black triangle-marked). with best configuration of the control gains (red circle-marked, ki = 0.1, kv = -0.18) and with worst configuration of the control gains (blue square-marked, ki = 0.3, kv = 0.45).

with regulation both with optimal values as from (5) (red line) and with worst values of control gains.

The optimal values for the control gains ki and kv-those related to the red circle-marked curve of Fig. 7-are ki = 0.1 and kv = -0.18. The stability analysis by means of the continuation method has been carried out and the results are reported in Fig. 6c and Fig. 6d. As for the one constant­power load it can be clearly noticed that the introduction of the control moves the Hopf bifurcation point towards the cusp of the nose-shaped curves.

IV. SIMULATION RESULTS

In order to evaluate the validity of the proposed theoretical approach in Section III, some results from time domain simulations are reported. In particular, the effectiveness of the pro-posed strategy in stabilizing the DC link dynamics (Fig. 2), is evaluated by comparing the state variables trends with and without the proposed control strategy. The synopsis of the framework in which the simulations have been carried out is reported in Table II.

The reference design, "case Oa", is characterized by the parameters reported in Table I. In particular the total power P requested by the loads is equal to 0.5 pu until to = 0.4 s,

when the total power is suddenly increased to 0.7 pu. The two loads are controlled in such a way that each load

has a power demand of 50% of the total power:

{PI = kP = 0.5P P2 = (1 - k) P = 0.5P

(8)

In Fig. 8 and Fig. 9 are reported the trends of the variations of the state variables of (2) (i.e., inductors currents and capacitors voltages) with respect to the steady state conditions.

TABLE II SYNOPSIS OF THE PERFORMED SIMULATIONS. to IS THE TIME AT WHICH THE STEP POWER INCREA SE b.P TAKES PLACE, T IS THE END TIME OF

SIMULATIONS, pO IS THE TOTA L POWER REQUEST OF THE LOADS BEFORE to. EACH POWER INCREA SE HAS BEEN TESTED WITHOUT A ND WITH THE

REGULATION.

name

case Oa

case Ob

case la

case Ib

0.2

0.15

0.1

0.05

-0.1 I , I

-0.15 I

" " , , , , , ,

to T

I :: I �: I regulation I s s

0.40 0.46 0.5 0.20 no

yes

0.40 0.46 0.5 0.68 no

yes

-0.25':. '--'0:-". '"',,--:o� . ''"''2--:-0-':. '::-3 -::-0.-':-":--- 0:-".7:",------:-'0.46 tis)

Figure 8. Variations of the state variables iI, current on inductor Ll, and VI, voltage on the capacitor Cb with respect to the steady state conditions in the reference "case Oa" with a total power P = 0.5 pu for t < to s and P = 0.7 pu for t � to s, k = 0.5.

0.2

0.15

0.1

-0.1

-0.15

-0.2

-0.25 '0. '--'0:-". '"'''--:0'''. ''''2--;:CO-:. ,""3 -""0.-':-,,:---' 0:-". ""5'------:-'0.46 tis)

Figure 9. Variations of the state variables i2 , current on inductor L2 , and V2 , voltage on the capacitor C2 ,with respect to the steady state conditions in the reference "case Oa" with a total power P = 0.5 pu for t < to s and P = 0.7 pu for t � to s, k = 0.5.

At to = 0.4 s, due to the step vanatlOn in the total power requested by the loads, a dumped oscillation starts until the new steady state condition is reached. In this transient the voltage variations on the two capacitors are below 5%, and the oscillations last for 500 ms.

From the results of the analysis shown in Fig. 7 it can be determined that the power limit that avoids the collapse of the whole system, due to the step variation in the power request, is equal to 1.18 pu. In fact, if such a sudden step variation happens, "case la", as shown in Fig. 10, the oscillations is not dumped and the system behavior is unstable. In addition it can be noticed that in any case the voltage and current oscillations

852

are too high.

1.5

" " " " " " ' , " " , , " " , . , I , I I I

, ' , I I , , I , I 0.5 , I , I I , I

I I I

I I I "

-1 .'

-1.5 0.' 0.41 0.42 0.43 0.44 0.45 0.46

t [s[

Figure 10. Variations of the state variables iI, current on inductor Ll, and VI, voltage on the capacitor C1 , with respect to the steady state conditions in the reference "case la" with a total power P = 0.5 pu for t < to s and P = 1.18 pu for t � to s, k = 0.5.

In order to prove the suitability of the proposed control strategy, the same step variation from 0.5 pu to 1.18 pu has been applied to the two constant-power loads system with the control feedback path described in (4). The values of the gains kv and ki respectively equal to -0.18 and 0.1, the optimal values obtained with the approach described in (5). In Fig. 11 and Fig. 12 trends of the variations of the state variables referred to the "case 1 b".

0.'

" , . 0.2 , I

, . "

-0.2 ,

-0.4 ,

-0.6 , , ,

-o.s'

-1 0.' 0.41 0.42 0.43 0.44 0.45 0.46

t [s[

Figure II. Variations of the state variables iI, current on inductor Ll, and VI, voltage on the capacitor Cl, with respect to the steady state conditions in the reference "case lb" with a total power P = 0.5 pu for t < to s and P = 1.18 pu for t � to s, k = 0.5, ki = 0.1, kv = -0.18.

From the analysis of the data reported in Fig. 11 and Fig. 12, the effectiveness of the proposed control strategy in improving the system stability is clear. In particular, in addition to the ability of the control strategy to avoid instability, it can be seen how the new steady state condition is reached in a shorter time, 250 ms, and with a lower energy content of the oscillations.

By analyzing Fig. 13, in which the voltage profile Vi of the reference voltage of the equivalent generator is reported during the transient, it can be seen how the control, as soon as the total power requested increases at to, forces a quick reduction of the controlled voltage Vi, in order to reach in the shortest time the new steady state condition.

Moreover, the control strategy lets the limit of the maximum power the loads can draw from the DC bus without losing

0.'

I -0.4 I

-0.6

-0.8

-6 '-0. ,--00-'0. ,C-, ----coe". '�2 -�o .� 43-�O"c. 44�� 0� . '�5 - o�."

tis]

Figure 12. Variations of the state variables i2, current on inductor L2, and V2 , voltage on the capacitor C2 ,with respect to the steady state conditions in the reference "case Ib" with a total power P = 0.5 pu for t < to s and P = 1.18 pu for t ;::: to s, k = 0.5, ki = 0.1, kv = -0.18.

1.16 r--�-�-�-�-�-�

1.14

1.12

1.1

1.08

:; 1.06

> 1.04

1.02

0.98

0.41 0.42 0.43 0.44 0.45 0.46 tis)

Figure 13. Control variable Vi of the equivalent generator in the time domain in the simulated "case Ib".

stability stay always above the nominal power for which the system has been designed.

Finally in order to evaluate from a quantitative point of view the quality of the control strategy in stabilizing the system when a power step variation occurs, "case Oa" and "case Ob", i.e., the same step variation in power request but, respectively, without and with regulation, have been simulated The energy variations associated to the oscillations of vi and i 1 with respect to the steady state conditions have been measured according to

iT Vi (i1 - in dH to

iT ( ( -kii1 - kvvI) i1 + to (9)

(kiir + kvvr) ir) dt, regulator on

iT Vi (i1 - in dt, regulator off to

where v� and i� are the steady state values for voltage of capacitor C1 and current of inductor L1. The results obtained from these simulations are shown in Fig. 14. In can be clearly noticed that the energy related to the oscillations when the regulator is active is less than (almost one half of) the energy that the system needs receive through the interface converter

853

to reach the stable steady state conditions when the regulator is disabled.

, ----------------------

o L---o�____c�-�-����� 0.4 0.41 0.42 0.43 0.44 0.45 0.46

tis)

Figure 14. Energy variations associated to the oscillations of Vi and il with respect to the steady state conditions for "case Oa" (blue solid line) and "case Ob" (read dashed line).

V. CONCLUSION

The paper has presented the stability analysis of a DC distribution system feeding constant power loads (CPLs). This analysis has been carried out using the numerical continuation method, which enables the study of nonlinear dynamical systems stability with respect to one or more parameters.

In fact, the intrinsic nonlinear behavior of CPLs may provoke unstable dynamical responses of the DC distribution system thus leading to collapse.

Two different configurations has been analyzed: i) one CPL; ii) one CPL directly connected to the DC bus and one CPL connected to the DC bus through a cable and a DC breaker. The introduction of this breaker makes it necessary to add a limiting inductor which contributes to modify the stability properties of the whole system.

The study of the second configuration-that with two CPLs-has been carried out with different values of the power share between the two loads, i.e., starting from all power requested by the farthest CPL to all power requested by the nearest CPL. Every scenario had different maximum power limits above which stability is lost. A feedback control path, that acts on input voltage and is proportional to DC bus volt­age and current, has been added to enhance stability. The optimal design of the regulator parameters has been also ad­dressed. According to the proposed approach, the controlled­system stability has been assessed and the best values of the control gains has been found.

ACKNOWLEDG MENT

This work has been carried out under the grant "MVDC Large Ship" within the POR FESR Program of the Regional Government of Friuli Venezia Giulia (Italy).

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